"TBA"

"Various topological and geometrical aspects of Yang Mills fields and their uses will be discussed."

"At high enough baryon densities, when the nucleons are crushed into quark matter, the quark matter that results is expected to be in one of a family of superconducting phases at low temperatures. Such kind of matter could be there in the core of neutron stars. Neutron stars are also expected to have a strong magnetic field. There has been a lot of progress during the last few years on the properties of quark matter in the presence of magnetic field. In the present talk, I shall discuss the current status of color superconducting phase of quark matter in the presence of strong magnetic field. Imposition of charge neutrality conditions here leads to 'gapless' modes. Possible consequences of such modes on transport properties of quark matter will be discussed."

"In this thesis we study the problem of deriving upper bounds on the positiveness of real polynomials. Such bounds have applications in algorithms for root isolation and approximation. Hence, bounds on positiveness need to efficiently computable. In the first part of the thesis, we focus on a well known bound for univariate polynomials due to Hong and a subsequent improvement over it due to Collins. Specifically, we show that unlike the Hong's bound, Collin's bound does not admit a linear time algorithm. In the multivariate setting, we derive a bound which improves upon the best known bound due to Hong. Our improved bound is derived by generalising a well known bound for univariate real polynomials due to Lagrange. Then, we give an algorithm to compute our improved bound and the running time of our algorithm matches the running time of the algorithm to compute Hong's bound. Our algorithm uses range trees for range querying. This motivates the question of how efficiently can we do range querying? Specifically, we focus on the tradeoff between storage and output complexity of data structures for range querying. In the second part, we show that a lower bound on the tradeoff between storage and output complexity due to Fredman can be strengthened. Our result implies that range trees are near optimal for range querying."